Direct solar absorption and storage in a molten glass thermocline

At high temperature, the effective thermal conductivity of molten glass becomes dominated by its transparency in the UV-to-2-micron passband. In the previous post it was calculated that molten glass at 2000 degrees K, with a 1/e (i.e., -4.3 dB) absorption length of 1 meter in the UV-to-2-micron passband (a loss rate of 4300 dB/km) would have an effective thermal conductivity of about 456 W/m-°K. The correlations for molten glass reported by Laurent Pilon et al., summarized in this chart, indicate an effective thermal conductivity of molten soda-lime glass at 2000 °K of 137 W/m-°K. From that value it can be estimated that the 1/e absorption length of soda-lime glass of commercial purity is about 137/456 m = 0.3 m. That is a figure that comports with everyday experience (e.g., looking through a glass panel edgewise.)

Thermophysical properties of molten glass at high temperature, including effective thermal conductivity, calculated from correlations in Laurent Pilon et al.

Absorption in the UV-to-2-micron passband of glass is almost entirely due to metal ion impurities, mainly iron. Thus improving the purity of ordinary glass by a factor of ten would extend the absorption length to 3 meters and increase effective thermal conductivity to about 1400 W/m-°K—about three times the room-temperature thermal conductivity of silver.

High effective conductivity also suppresses natural convection since natural convection starts out as a wiggle in an isotherm that enlarges by gravitational instability—despite viscosity slowing it down—faster than it can be damped out by thermal diffusion.

Solar radiation is 90% within the UV-to-2-micron passband of glass, so it plays by the same rules as the conductivity-dominating thermal radiation. A net flux of say, 0.4 MW/m2, onto the top of the glass melt will not penetrate any more deeply if it is the result of solar radiation or thermal radiation (aka, effective conduction.) Sunlight has relatively more short wavelength energy, of course, but under the assumption of spectrally uniform absorption in the UV-to-2-micron band, this makes no difference.

Flux is flux. That greatly simplifies modeling.

Solar heating of a glass melt from above stores heat in a stable thermocline. Later radiant cooling of the melt from above "excavates" the stable thermocline removing heat by natural convection from successively deeper and deeper levels until the whole melt is at the "empty" temperature. Temperatures in the diagram are measured relative to empty. Modeled in Energy2D.  Depth 10 m, absorption length 10 m, shown after 6 hours of solar heating.

Effective thermal conductivity of high-purity glass melts

The effective conductivity of high-purity glass melts increases rapidly with temperature because the melt is semi-transparent to its own thermal radiation. At wavelengths shorter than 2 microns a high purity glass melt may permit its own thermal photons to travel some meters, in some cases even tens of meters, before being re-absorbed. At temperatures used for solar thermal storage, 1840 °K to 2070 °K, about a quarter to a third of blackbody radiation is in the below-2-micron passband of glass. Since effective thermal conductivity is proportional to the average distance a blackbody photon travels before it is reabsorbed, the effective conductivity of a glass melt is a function of its level of purity.



At absolute temperature T, blackbody radiation is

σT⁴,

where σ is the Stefan-Boltzmann constant,

σ = 5.7 E−8 W m−2 K−4 .

The derivative of blackbody flux with respect to T is:

 4σT³ W/m2-°K,

 which is the transfer between two blackbody surfaces differing slightly in temperature. At glass-melt storage temperatures only about a quarter of the blackbody radiation is in the below-2-micron passband of the melt, so we are only concerned with a differential flux of about

σT³ W/m2-°K.

If the average temperature is 2000 °K and photons in the passband travel a distance of one meter, the effective thermal conductivity (ignoring actual phonic conduction which will be relatively small) is

(5.7 E−8 W m−2 K−4)(2000 °K)³(1 m) = 456 W/m-°K

Compare the thermal conductivity of silver at room temperature is 429 W/m-°K.

Radiative view factors in glass-melt solar storage

Radiative view factors between a cylinder and its endcaps. Diagram from Isidoro Martinez, "Radiative View Factors."

In radiative heat transfer, a view factor F_A→B is the proportion of the radiation which leaves surface A and (directly) strikes surface B. In some cases there may also be an adiabatic blackbody surface C which, at thermal equilibrium, must re-radiate all of the photons it receives. In this case we can define an effective view factor  F*_A→B that includes the re-radiated photons that reach B indirectly:

F*_A→B = F_A→B + (F_A→C)(F_C→B)

When all the surfaces are blackbodies (an assumption that greatly simplifies the math, and is approximately true in practice) the net radiative flux from A to B, averaged over  S_A, the area of A, is:

Φ_A = F*_A→B * σ(T_A⁴ - T_B⁴),

where σ is the Stefan–Boltzmann constant, σ = 5.7 E−8 W m−2 K−4.

The corresponding net flux at surface B, Φ_B, must be Φ_A multiplied by the ratio of the two areas:

Φ_B = Φ_A * (S_A/S_B) = F*_A→B * σ(T_A⁴ - T_B⁴) * (S_A/S_B).

When solar energy is stored in a glass melt, we can consider the free surface of the melt to be surface A (the lower cap of a cylinder,) the water wall tubes of the boiler to be surface B (the walls of the cylinder,) and the roof of the furnace to be the adiabatic surface C (the upper cap of the cylinder.)

Water wall tubes. Image quoted from http://tubeweld.com.

In the view factor diagram above, a reduced radius 'r' is related to the cylinder's radius, R, and its height, H, by:

r = R/H,

and a parameter ρ is defined to be:

ρ = (√(4 * r2 + 1) -1) / r.

The correspondences between our lettered surfaces and the diagram's numbered surfaces are:

A = 3, B = 1, C = 4.

From the diagram,

F_A→B = F_3→1 = ρ/2r

F_A→C = F_3→4 = 1 - ρ/2r

F_C→B = F_4→1 = F_3→1 = ρ/2r

So,

F*_A→B = F_A→B + (F_A→C)(F_C→B) =  ρ/2r + (1 - ρ/2r)(ρ/2r) = (ρ/2r) (2 - ρ/2r),

and the area ratio, S_A/S_B, is

S_A/S_B = πR² / 2πRH = R/2H = r/2.

The average flux on the water wall, Φ_B, is

Φ_B = F*_A→B * σ(T_A⁴ - T_B⁴) * (S_A/S_B) = (ρ/2r) (2 - ρ/2r) (r/2) * σ(T_A⁴ - T_B⁴)

Φ_B = (ρ/4) (2 - ρ/2r) * σ(T_A⁴ - T_B⁴)

With R = 51 m, H = 87 m, r = 0.58, ρ = 0.93, T_A = 1610 °K (1340 °C) , T_B = 730 °K (460 °C), as calculated in an earlier post,

Φ_B = (ρ/4) (2 - ρ/2r) * σ(T_A⁴ - T_B⁴) = 0.26 * 367 kw/m2 = 96 kw / m2

which is not nearly the 250 kW/m2 we need to see on a water tube wall.

We need to increase radiant transfer by upping the "empty" temperature of the glass melt. This will also incidentally decrease storage density, resulting in an increase in R and a decrease in H (as calculated on the basis of 250 kW/m2.)

A spreadsheet exploring radiant transfer in glass-melt thermal storage.

Exploring these relations in a Numbers spreadsheet shows that Tempty = 1570 °C gives a consistent solution with R = 63 m, H = 71, and the flux on the water wall tubes = 250 kw/m2. The glass temperature range from empty to full is just 230 °C.